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Quasi-Newton inversion of seismic first arrivals using source finite
bandwidth assumption: application to landslides characterization
from velocity and attenuation fields.
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Julien Gance (1,2), Gilles Grandjean (1), Kévin Samyn (1) and Jean-Philippe Malet (2)
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Keywords:
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Inverse theory, Seismic tomography, Fresnel wavepaths, Finite frequency, Resolution improvement,
Landslide
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(1) BRGM, Bureau de Recherches Géologiques et Minières, 3 Avenue Claude Guillemin, F-45060
Orléans, France
(2) Institut de Physique du Globe de Strasbourg, CNRS UM7516, EOST/Université de Strasbourg, 5
rue Descartes, 67084 Strasbourg Cedex, France.
Corresponding author:
Julien GANCE
BRGM, Natural Hazard Department.
3 Avenue Claude Guillemin BP36009 45060 Orléans Cedex 2, France
Phone : +33 (0)2 38 64 38 19
Fax : +33 (0)2 38 64 35 49
E-mail address: j.gance@brgm.fr
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1.
Introduction
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Near-surface geophysical techniques are of great potential to image subsurface structures in many
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geomorphological studies (Schrott and Sass, 2008; van Dam, 2010), and even more specifically for
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improving the knowledge on landslide features (Jongmans and Garambois, 2007). Landslide analysis
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generally involves the combined use of several geophysical methods to image different petrophysical
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parameters (Bruno and Marillier, 2000; Mauritsch et al., 2000; Méric et al., 2011). Seismic surveys are
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particularly well adapted to detect and characterize specific features of landslides such as spatial
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variation in compaction and rheology of the material (Grandjean et al., 2007), variation in fissure
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density at the surface (Grandjean et al., 2012), and complex slip surface geometries (Grandjean et al.,
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2006). As the wave propagation is mainly controlled by the elastic properties of the medium, seismic
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surveys are often well correlated with geotechnical observations.
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Among them, different approaches can be used for processing data depending on the analysis of
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the different kind of waves associated to particular propagation phenomena. Bichler et al. (2004) and
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Bruno and Marillier (2000) interpreted the late arrivals of P-waves to image the bedrock geometry
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using seismic reflection analysis. Mauristch et al. (2000) and Glade et al. (2005) analyzed refracted
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waves to define the internal geometry (e.g. layering) of the landslides. Grandjean et al. (2007) and
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Grandjean et al. (2011) studied the first arrival traveltimes to recover P-wave velocity distribution
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across landslides. More recently, Samyn et al. (2011) used a 3D seismic refraction traveltime
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tomography to provide a valuable and continuous representation of the 3D structure and geology of a
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landslide. Grandjean et al. (2007) and Grandjean et al. (2011) used also Spectral Analysis of Surface
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Waves (SASW) techniques to obtain S-wave velocity distributions along landslide cross-sections.
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Finally, based on the work of Pratt (1999) and Virieux and Operto (2009), Romdhane et al. (2011)
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demonstrated the possibility to exploit the entire wave signal by performing a Full elastic Waveform
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Inversion (FWI) on a dataset acquired on a clayey landslide.
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These methods are based on more or less important assumptions on the soil type and their
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petrophysical characteristics, on the complexity of layers, and on their geometry, and are therefore not
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recommended for all case studies. Seismic refraction can be basically performed using the General
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Reciprocal Method (GRM; Palmer and Burke-Kenneth, 1980) or the Plus/Minus Method (Hagedoorn,
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1959). These methods are fast and easy to implement, but mostly based on simple hypotheses - such as
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a layered media - and are consequently not very efficient to recover important variations in lateral
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velocity. Reflection surveys need to record high frequency waves to obtain a good resolution of final
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stacked sections; they are difficult to set up for subsurface applications because of the strong
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attenuation affecting high frequencies and decreasing the signal to noise ratio (Jongmans and
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Garambois, 2007). Moreover, this method produces a reflection image that cannot be directly and
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easily interpreted as direct information on the mechanical properties of the medium that could be
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derived from velocity fields. FWI appears to be the more advanced method since it is based on the
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realistic assumption of an elastic media and uses the whole seismic signal in the inversion scheme.
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However, it remains a complex method, requiring an important data pre-processing (source inversion,
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amplitude correction, initial model estimation) that is hardly applicable to near surface real data
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featured with a low signal to noise ratio and complex waves interactions (Romdhane et al., 2011). As a
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consequence, the issue of recovering the structural image of a landslide from the seismic velocity field
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estimated with an accurate method is still a challenge.
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The objective of this work is to refine the first arrival tomography approach applied to landslide
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analysis, which is a good compromise between the strong assumptions made in simple refraction
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methods and the complexity of the FWI approach when used in very heterogeneous soils. P-wave
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tomography is largely used, from crustal to shallow geotechnical studies and allows recovering, at a
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relatively low cost, a reliable image of the velocity distribution. In this perspective, Taillandier et al.
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(2009) proposed to invert first arrival traveltime tomography using the adjoint state method, but had to
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face the problem of gradient regularization. The proposed method is based on a Hessian formulation
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(Tarantola, 1987) to ensure an optimum convergence of the velocity model during iterations. We only
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use here the first arrivals of the seismic signal caused by direct or refracted waves. In that case, late
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arrival, including scattering, conversion effect are not taken into account in the inversion.
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In the next section, we detail the theory used to formulate the inversion equation and the related
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approximations introduced for optimizing the algorithm and ensuring the convergence toward the best
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solution. The final algorithm is tested on a synthetic case in order to estimate the quality of the
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velocity field reconstruction without considering noisy data. To compare our results with a reference
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case, the synthetic velocity model is identical to the one used by Romdhane et al. (2011). Then, we
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process a real seismic dataset obtained from a seismic survey at the Super-Sauze landslide (South
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French Alps) in clayey material. On this case study, we invert for the P-wave velocity field but also for
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the attenuation factor. The effects of noisy data on the quality of the reconstructed velocity field are
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analyzed, and the observed surface variation in the velocity field is discussed by integrating other
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sources of information (geotechnical tests and geomorphological interpretation).
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2.
Theoretical approach
a.
General inverse problem
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Solving the inverse problem requires a modeling step (e.g. the direct problem) for computing the
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residuals (e.g. the difference between computed and observed data). An updated model can be
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estimated by back-projecting the data residuals on each cell of the model discrete grid. The direct
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problem thus consists in finding a relation between the data t, taken here as the first arrival traveltimes,
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and the physical P-wave velocity model s, such as:
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𝑡 = 𝑓(𝑠)
(1)
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This problem is generally solved using the asymptotic high frequency wave propagation
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assumption in a non-homogeneous isotropic medium. The wave propagation equation is then
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simplified into the eikonal equation that computes the first arrival traveltimes over a discrete grid for
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example by using a finite-difference scheme (Vidale, 1998). This technique has been improved for
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sharp velocity contrasts (Podvin and Lecomte, 1991), optimized by Popovici and Sethian (1998) who
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proposed the Fast Marching Method (FMM), by Zhao (2005) who proposed the Fast Sweeping
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Method, and recently adapted to non-uniform grids (Sun et al., 2011). The technique is actually largely
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used because of its efficiency and accuracy. In this study, we use the FMM algorithm (Popovici and
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Sethian, 1998) to compute the direct problem. The computation of the first arrival traveltime is
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generally followed by a ray tracing that uses the traveltimes maps to propagate the wavepath from the
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sources to the receivers. The traveltime t can then be expressed using the following matrix form:
𝑡 = 𝐿𝑠
(2)
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where L is the length of the ray segment crossing each cell of the slowness model S. Generally, the
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Fermat principle allows the Fréchet derivatives matrix to be approximated by the matrix 𝐿 (Baina,
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1998). This principle establishes that a small perturbation 𝛿𝑠 on the initial model does not change the
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ray paths, and therefore, only affect the traveltime 𝛿𝑡 at the second order. This linearization of the
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problem around the slowness model 𝑠 𝑘 implements the Quasi-Newton (Q-N) assumption and makes
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the resolution of the nonlinear problem to be solved by the following tomographic linear system:
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[(𝐿𝑘 )𝑇 𝐿𝑘 ]𝛿𝑠 = [(𝐿𝑘 )𝑇 𝛿𝑡]
(3)
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where k and T represent respectively the iteration index and the transpose operator. Several
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techniques can be used to solve Eq. 3. The Simultaneous Iterative Reconstruction Technique (SIRT) is
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the most common because it does not require a large computer memory; consequently, SIRT is widely
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used to invert large sparse linear systems (Trampert and Leveque, 1990). Its convergence toward a
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least-square solution has been proved (Van der Sluis and Van der Vorst, 1987) but this method suffers
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from few drawbacks: SIRT introduces an intrinsic renormalization of the tomographic linear system
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that modifies slightly the final solution. For that reason, we propose to use in our approach the LSQR
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(Least-Square QR) algorithm (Paige and Saunders, 1982) that has been proved to be superior to SIRT
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or Algebraic Reconstruction Techniques ART algorithms in terms of numerical stability and
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convergence rates (Nolet, 1985).
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b.
From rays to Fresnel volumes
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Revisiting seismic traveltimes tomography also needs to consider the ray approximation, e.g. the
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infinite spectral bandwidth assumption. The main issue of this approximation lies in the traveltime
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computation taken as line integrals along the rays spreading over the slowness model. Because the
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slowness values are only considered along ray paths, the problem is often underdetermined and leads
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to numerical instability (Baina, 1998). In practice, this difficulty is generally by-passed using
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regularization operators to reduce the non-constrained part of the model, and then, to reinforce the
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numerical stability. Generally, a simple smoothing of the reconstructed slowness model (Zelt and
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Barton, 1998), the application of a low-pass filter on the gradient (Taillandier, 2009) or the elimination
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of the lowest eigenvalues in the Hessian matrix (Tarantola, 1987) can be used. However, such
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regularization operators require the selection of appropriate parameters (filter length, eigenvalue cutoff
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and more generally the size and weights of the smoothing operators). Kissling et al. (2001)
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demonstrated the dependency of these parameters on the resolution and the quality of the final model
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in poor wavepath coverage areas. This is the reason why the use of physically-based regularization
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operators such as Fresnel volumes or sensitivity kernels is preferable since they are based on non-
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subjective principles completely defined by the problem.
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With this assumption, several methods have been developed including those using the concept of
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Fresnel volumes as a regularization factor. Nolet (1987) proposed to use the size of the Fresnel zone to
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constrain the size of a spatial smoothing operator. Vasco et al. (1995), Watanabe et al. (1999),
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Grandjean and Sage (2004) or Ceverny (2001) use the Fresnel volume in the back projection of the
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residuals.
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More recently, the concept of sensitivity kernels or Fréchet Kernel introduced by Tarantola (1987)
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has been reformulated for traveltime tomography. It represents a good compromise between the strong
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assumption of the asymptotic ray theory and the high computational cost of the complete full wave
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inversion tomography (Liu et al., 2009). While ray theory is well adapted in media characterized by
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geological structures larger than the first Fresnel zone, the use of sensitivity kernels allows to
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overcome this constraint and then, to increase the spatial resolution (Spetzler and Snierder, 2004).
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Several authors have developed this concept and investigated the properties of the sensitivity kernels
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for traveltime tomography in homogeneous media, mainly at the spatial scale of the earth crust
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(Dahlen et al., 2000; Dahlen, 2005; Zhao et al.; 2006). The strategies proposed by these authors are
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generally limited to smoothly heterogeneous media (Liu et al., 2009; Spetzler et al., 2008) and are
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barely applied to real datasets, mainly because of prohibitive calculation costs (Liu et al., 2009).
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Sensitivity kernels and Fresnel volumes are constructed on the single scattering (Born)
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approximation. This approximation considers that a part of the wave can be delayed by velocity
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perturbation in the vicinity of the ray path. For highly heterogeneous media, this approximation is not
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valid anymore because multiple scattering should be taken into account to fully explain observed first
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arrivals.
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In the particular case of subsurface soil investigation where highly heterogeneous media are
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observed, the use of sensitivity kernels is not recommended since it would involve complex algorithms
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and computing times similar to those featuring the FWI approach. In this context, we decided to solve
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empirically the problem of complex multiple scattering inside the first Fresnel zone, this approach
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allowing to address issues of regularization by using increasing finite frequencies bandwidths.
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Therefore, the developed algorithm assumes that high spatial heterogeneities of the soil around the
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Fresnel wavepath affect, through complex multiple scattering, the first arrivals of the signal.
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In the next section, we explain how the Fresnel volumes are defined as a simplification of the
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sensitivity kernels and implemented in a Q-N algorithm.
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We first express the Fresnel weights proposed by Watanabe et al. (1999) and used by Grandjean and
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Sage (2004), in order to materialize the wavepath in the model. They classically decrease linearly from
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a value of one (when the cell is positioned on the ray path) to a value of zero (when it is out of the
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Fresnel volume):
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1 − 2𝑓Δ𝑡,
𝜔=
{
0,
0 ≤ Δ𝑡 <
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≤ Δ𝑡
2𝑓
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2𝑓
(4)
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Watanabe et al. (1999) and Grandjean and Sage (2004) defined these weights ω as the probability
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that a slowness perturbation delay the arrival of the wave by a Δt, where f is the considered frequency
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of the wave. This definition is interesting because of its simple expression that allows fast computing
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while taking into account the global shape of the 2D traveltime sensitivity kernel given of Spetzler and
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Snieder (2004), corresponding to a decrease of sensitivity until the first Fresnel zone.
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Because the size of the Fresnel volume is depending on the considered source frequency, a new
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inversion strategy based on increasing frequency can be proposed. To consider the entire source
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frequency content, some authors compute wavepaths in a band-limited sensitivity kernel, stacking the
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monochromatic kernels with a weight function similar to the amplitude spectrum of the wavelet
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(Spetzler and Snieder, 2004; Liu et al., 2009). From our side, we propose to compute the Fresnel
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weights for a monochromatic wave, increasing its frequency at each step of the inversion. The
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considered frequency sampling with an increasing rule ranging from the lower to the higher frequency
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of the source signal. We also choose to give the same weight to all frequencies in order to limit the
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number of iteration and to preserve the rapid convergence of the algorithm. Taillandier et al. (2009)
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showed that this method, applied on the gradient filtering permit to overcome the non-linearity of the
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problem. For low frequency values, the Fresnel zone will be wider and the slowness model will be
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reconstructed with large wavelengths. Conversely, for high frequency values, the Fresnel zone will be
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thinner and the slowness model will be reconstructed with sharp wavelengths. This strategy appears
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efficient to improve the resolution during the inversion - and so the convergence of the algorithm - in
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full wave inversion where it prevents from cycle-skipping issues (Sirgue and Pratt, 2004; Virieux and
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Operto, 2009; Romdhane, 2011). Recovering slowness variations whose sizes are in agreement with
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each wavelength transmitted by the source should lead to better images of local heterogeneities while
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preserving the algorithm convergence.
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With these assumptions, the traveltime perturbation can be expressed as the integral over a Fresnel
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volume multiplied by the slowness perturbation field observed for all points r in the volume (Liu et al,
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2009). This linear relationship among traveltime and slowness perturbation is the result of the first
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Born approximation that is only valid for small perturbation (Yomogida, 1992):
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𝛿𝑡 = ∫ 𝑊(𝑟)𝛿𝑠(𝑟)𝑑𝑟
(5)
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where 𝑊 is the Fresnel weight operator (Fig. 1) normalized by the area of the Fresnel ellipse
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perpendicular to the raypath 𝑎, for each shot and receiver such as:
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𝑊(𝑟) =
1
𝜔(𝑟)
𝑎
(6)
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This normalization allows linking our approach to the geometrical ray theory in the case of a plane
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wave propagation (Vasco et al, 1995; Spetzler and Snieder, 2004; Liu et al, 2009), so that the integral
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of W over a surface perpendicular to the wavepath is equal to one, which is generally verified for
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sensitivity kernels:
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+∞
∫
𝑊(𝑠, 𝑟, 𝑓)𝑑𝑠 = 1
(7)
−∞
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The Fresnel volume thus defined, also called Fréchet kernel corresponds to the Fréchet derivative
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(Yomogida, 1992; Tarantola, 1987) or Jacobian matrix of the forward problem, similar to the length of
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ray segment matrix 𝐿 defined in the asymptotic ray theory. The problem, considered in 2D, can then
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be written in its matrix form (Vasco et al, 1995):
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𝛿𝑡 = 𝑊 𝛿𝑠
(8)
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where 𝑊 is the [number of data x number of cells] Fresnel weight matrix, calculated for each shot and
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for each receiver such as for the cell j and for the couple source-receiver i:
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𝑊𝑖𝑗 =
𝜔𝑖𝑗
𝑙
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where 𝑙 is the length of the Fresnel surface along a direction perpendicular to the ray path.
(9)
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We are however aware that the use of the Fresnel weight as defined previously is a rough
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approximation of the wavepath. This is the reason why we prefer to keep the ray assumption to
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compute traveltimes, avoiding the introduction of any error due to this approximation. This choice has
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been justified by Liu et al. (2009) who observed small differences between traveltime values computed
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with both approaches.
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c. New implementation of the inverse Problem
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In the previous section, we showed that the Born approximation combined with Fresnel volumes
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allowed the estimation of Fréchet derivative. To solve the nonlinear inverse problem, we use a steepest
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descent iterative algorithm to minimize the 𝐿2 norm misfit function l (Tarantola, 1987) that can be
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written in this particular case:
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𝑙(𝑠) =
1
𝑇
[(𝑓(𝑠) − 𝑡𝑜𝑏𝑠 )𝑇 𝐶𝑇−1 (𝑓(𝑠) − 𝑡𝑜𝑏𝑠 ) + (𝑠 − 𝑠𝑝𝑟𝑖𝑜𝑟 ) 𝐶𝑆−1 (𝑠 − 𝑠𝑝𝑟𝑖𝑜𝑟 )]
2
(10)
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where 𝐶𝑇 and 𝐶𝑆 are respectively the covariance operators on data and model, 𝑓 represents the
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theoretical relationship between the model 𝑠 and the traveltimes 𝑡, 𝑡𝑜𝑏𝑠 the observed data and 𝑠𝑝𝑟𝑖𝑜𝑟
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the a priori information on the model.
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The Q-N method consists in minimizing the misfit function iteratively using its gradient and
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approximated Hessian matrix. The gradient gives the direction of steepest descent of the misfit
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function while the Hessian matrix is used as a metric indicating its curvature, approximated locally by
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a paraboloïd (Tarantola, 1987). Compared to the gradient method, the Hessian is here used to improve
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the assessment of direction and norm of the update vector that is applied to the slowness model, so that
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each step is performed in the good direction and with an optimized length. The tomographic linear
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system in the case of no a priori information on the model can be written in its matrix form:
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[(𝑊 𝑘 )𝑇 𝐶𝑇−1 (𝑊 𝑘 )] 𝛿𝑠 = [(𝑊 𝑘 )𝑇 𝛿𝑡]
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(11)
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This linear tomographic system can be solved by using a LSQR algorithm. The Fresnel weights
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matrix employed is a [number of data x number of cells] matrix. Its computation is straightforward on
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classical personal computers. The Q-N algorithm is based on the inverse of the Hessian matrix that is a
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square matrix of size [number of data x number of data] that can be difficult to invert for large dataset.
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In our case, we used cluster computing to invert our data and optimized the step size along the
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direction given by the Hessian using a scalar to weight the slowness update thanks to a parabolic
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interpolation (Tarantola, 1987).
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d. Validation on a synthetic dataset
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To compare the performance of our algorithm with full wave inversion, it was tested on the synthetic
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transverse section used by Romdhane (2011) representing a typical cross-section of a landslide (Fig.
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2a). The performance of the method is discussed by taking as a reference the SIRT algorithm of
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Grandjean and Sage (2004).
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The synthetic dataset has been calculated with a simple 2D eikonal equation solver, so that traveltimes
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are not perturbed by scattering effects. The dataset is composed of 50 shots recorded on 100
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geophones. The model consists in 209x68 cells of 1 m in width. The seismic sources are spaced
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regularly every 2 m and the geophones every 1 m. The synthetic model is composed of ten layers with
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P-wave velocity ranging from 110 m.s-1 to 3300 m.s-1. It represents a transversal cross-section of the
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landslide body lying on a homogeneous consolidated bedrock. Except between the landslide body and
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the bedrock, the changes in P-wave velocity are small, so that it is difficult to reconstruct the detailed
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shape of the landslide body. The stopping criterion is a change in the cost function lower than 1% and
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the inversion is limited to 20 steps.
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Figure 2b and 2c compare the results obtained with the SIRT algorithm (Grandjean and Sage, 2004)
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and the Q-N one. They converge respectively in 15 and 20 iterations. The convergence of the misfit
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function is better for the Q-N algorithm with a value 3.6 times lower than the SIRT. Globally, the Q-N
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algorithm succeeds in recovering the shape of the bump at 2000 m.s-1 located on the left side of the
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cross-section and renders a more spatially detailed shape of the bedrock topography (Fig. 2b, 2c). The
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SIRT algorithm does not retrieve those initial structures. We can notice that both algorithms have
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difficulties to reproduce the low velocity of the very near surface layer, because of the poor ray
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coverage occurring in this area.
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Figure 2d, 2e and 2f compare the results on three vertical cross-sections chosen to highlight the ability
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of the algorithm to recover P-wave velocity with a better accuracy than the SIRT. We can notice that
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for the top-soil, P-wave velocity are the same for both algorithms and are equal to the P-wave
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velocities of the initial model in that zone. Moreover, The Q-N algorithm seems to be able to recover
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lowest wavelengths than the SIRT one as on figure 2d between 55 and 60 m of depth.
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This synthetic example demonstrates the ability of the Q-N algorithm to recover the shape of the
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internal layers of a landslide body. Contrary to the SIRT result, the Q-N allows to interpret correctly
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the thickness and geometry of the different layers.
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3.
Application of the Q-N algorithm on a real dataset
a. Study site presentation
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The Q-N algorithm has been applied on a real dataset acquired at the Super-Sauze landslide (South
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French Alps). The landslide has developed in Callovo-Oxfordian black marls. Its elevation is between
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2105 m at the crown which is established in in-situ black marls covered by moraine deposits and 1740
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m at the toe. The landslide is continuously active with displacement rates of 0.05 to 0.20 m.day-1
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(Malet et al., 2005a). The detachment of large blocks of marls from the main scarp (Fig. 3) and their
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progressive mechanical and chemical weathering in fine particles explain the strong grain size
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variability of the material especially in the topsoil (e.g. decametric blocks of marls at various stages of
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weathering, decimetric and centimetric clasts of marls, silty-clayed matrix; Maquaire et al., 2003). The
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bedrock geometry is complex with the presence, in depth, of a series of in-situ black marl crests and
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gullies, partially or totally filled with the landslide material (Flageollet et al., 2000; Travelletti and
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Malet, 2012). This complex bedrock geometry delimits sliding compartments of different
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hydrogeological, rheological and kinematical pathways. The variable displacement rates and the
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bedrock geometry also control the presence of large fissures at the surface (Fig. 3b, 3c) that can be
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imaged with joint electrical and electromagnetic methods (Schmutz et al., 2000). Many kinds of
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heterogeneities are observed at different scales, and they control directly the mechanical behavior of
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the landslide by creating excess pore water pressures (Van Asch et al., 2006; Travelletti and Malet,
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2012). The top soil surface characteristics have also a large influence on the surface hydraulic
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conductivity (Malet et al., 2003) and therefore, on the water infiltration processes (Malet et al., 2005b;
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Debieche et al., 2009). In this context, we tested the Q-N P-wave tomography inversion scheme to
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provide a high-resolution characterization of structures and detect small scale heterogeneities along a
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N-S transect of the landslide.
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b. P-wave velocities inversion
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The seismic profile is parallel to the main sliding direction of the material, and is located in the upper
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part from the secondary scarp to the middle part of the accumulation zone (Fig. 3). The base seismic
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device has a length of 94 m long and consisted of 48 vertical geophones (with a central frequency of
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10 Hz) regularly spaced every 2 m. The 60 shots have been achieved with a hammer every 4 m. We
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used a roll-along system to translate the acquisition cables with a 24 geophones overlap and then
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investigate a 238 m long linear profile. The recording length is 1.5 s with a sample rate of 0.25 ms and
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the acquisition central consists in a Geometrics Stratavizor seismic camera (48 channels). The shots
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show a relatively good signal to noise ratio until the last geophone in the upper part of the section. The
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shots of the lower part are affected by the higher density of soil fissures. The attenuation and the signal
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to noise ration are consequently higher in this part of the profile. The signal is dominated by the
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surface waves but the first arrivals are clearly visible for all seismic shots.
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To estimate the pick uncertainty, the time difference of the picks from reciprocal source-receiver pairs
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was examined. Two different people picked the first arrival of the waves, and for each created dataset
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we studied the difference between reciprocal traveltimes. From each dataset, we created 3 subsets: a
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first one kept intact, a second one where we corrected the picks having more than 10 ms of difference
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in traveltime reciprocity, the picks being removed when not possible and a third one removing picks
349
having a difference of traveltime reciprocity greater than 10 ms. We inverted the 6 datasets and kept
350
the one which gave the lower final misfit function value and the P-wave tomogram the most reliable
351
with our a priori knowledge of the studied zone. The differences between P-wave tomograms were not
352
so important, but the dataset selected permitted to interpret the results unequivocally. The error in
353
reciprocity of traveltimes can be approximated by a Gaussian belt with a standard deviation of 2.2 ms
354
(Fig. 5).
355
356
The inversion has been performed with the Q-N algorithm on a 358 x 153 grid. Each square cell of
357
the grid measures 0.67 m. The hammer source gave frequencies comprised between 30 Hz and 120 Hz
358
with a dominant frequency of 40 Hz. Those frequencies are also present in the P-wave up to the end of
359
the profile (Fig. 4b). Regarding the different amplitude spectra, we considered those values as constant
360
for each shot. For the first iteration, the frequency was set to 30 Hz and then increases at each step to
361
respectively 45 Hz, 60 Hz, 75 Hz, 90 Hz, 105 Hz and 120 Hz; after the eighth steps, the frequency was
362
kept constant at 120 Hz.
363
364
365
The initial model is of a first importance for a good convergence. Indeed, the linearization of the
366
problem around the initial model is only valid for model close to the real one. This reason motivates
367
the automated calculation of the initial model from the data by using a simplified slant-stack algorithm
368
transforming the dataset in the velocity - intercept times domain. Then, theoretical traveltime curves
369
were calculated in a double loop for a velocity v ranging from 𝑣𝑚𝑖𝑛 to 𝑣𝑚𝑎𝑥 (defined by the user) and
370
for a range of intercepts 𝜏 such as:
371
𝑡=𝜏+
372
𝑜𝑓𝑓𝑠𝑒𝑡
𝑣
(12)
373
For each of these theoretical curves, the model was compared to the observed traveltimes; the
374
number of points with a difference of less than 0.5 ms was stored in the 𝜏-p plan (Fig. 6a). For each
375
intercept, the velocity corresponding to the maximum number of point was used to create a profile of
376
velocity that is further used under the shot point (Fig. 6b).
377
The depth 𝑑 corresponding to each velocity was then deduced from the velocities by using Eq. 13:
378
𝑑(𝑖) = 𝑑(𝑖 − 1) + 𝑣(𝑖) ∗ 𝛿𝑡
(13)
379
380
where 𝛿𝑡 represents the sample rate of intercepts (Fig. 6c).
381
The velocity profiles were gathered to form the initial model that was finally smoothed to
382
avoid lateral artifacts. With a misfit function value of 0.3612, this initial model is supposed to be as
383
much as possible close to the final one which is the best way to avoid divergence or convergence in
384
local minimum. The stopping criterion was set to a percentage change lower than 1 % and the number
385
of iteration was limited to 20. The misfit function value associated to the final model is 0.0122 and has
386
been obtained after 17 iterations. Figure 7 shows the misfit functions related to the Q-N and SIRT
387
algorithms for comparison.
388
Figure 8 shows respectively the initial velocity model and the inverse models obtained with both SIRT
389
and Q-N algorithms.
390
391
c. Seismic wave attenuation tomography
392
393
The amplitude of the first arrival is exploited here to image the wave attenuation. Seismic wave
394
attenuation is an efficient physical property because it is directly linked to porosity and to the presence
395
of fissures in the media (Schön, 1976). Several attenuation parameters can be used to invert seismic
396
attenuation: spectral ratio, centroid frequency shift and peak frequency shift (De Castro Nunes et al.,
397
2011). In our analysis, we used the simple method proposed by Watanabe and Sassa (1996)
398
considering that in a homogeneous attenuation media the amplitude of the spherical wave verifies:
𝐴(𝑟) =
𝐴0 −𝛼𝑟
𝑒
𝑟
(14)
399
400
where 𝐴0 is the amplitude of the source signal, r the distance from the seismic source along the
401
raypath and α the attenuation coefficient.
402
Applying a logarithm function, the equation becomes linear in 𝛼. We proposed to compute the
403
attenuation tomography after the P-wave velocity field so that the problem is reduced to a simple
404
linear problem:
𝛼 = 𝛼 + 𝑊 𝑇 𝛿𝐴
(15)
405
406
where α is the attenuation coefficient, 𝛿𝐴, the amplitude update and W the Fresnel weights matrix.
407
We performed five iterations starting from a simple homogeneous media with an attenuation 𝛼 =
408
1.0 𝑒 −03 Np.m-1 and calculated the Fresnel weights matrix for the Q-N inverted Vp model and for five
409
frequencies (30 Hz, 45 Hz, 60 Hz, 90 Hz, 120 Hz). Although the results are generally presented
410
through the dimensionless quality factor, in our case, the dominant source frequency is almost constant
411
for each shot (40 Hz) and the raw results are more contrasted. Moreover, because the definition of the
412
quality factor can be different for lossy (highly attenuating) materials (Schön, 1976), we image
413
directly the attenuation map. In order to compare the attenuation map with surface fissuring, a
414
geomorphological inventory of the fissures was created by direct observations in the field along the
415
seismic profile. All the fissures of width larger than 0.05 m were mapped. The Surface Cracking Index
416
(SCI), defined as the total length of fissures per linear meter of the seismic profile, is represented in
417
Figure 9a.
418
419
420
421
4. Interpretation and discussion
422
The velocity model is firstly compared to the geometrical model proposed by Travelletti and Malet
423
(2012) proposed from the integration of multi-source data (e.g. electrical resistivity tomographies,
424
dynamic penetration tests and geomorphological observations) at coarser scale.
425
The bedrock depths obtained from penetration tests were used to deduce the minimum velocity of the
426
bedrock fixed at 850 m.s-1. The bedrock position thus obtained is compared to the one of Travelletti
427
and Malet (2012) along the profile (Fig. 10a, b). The thickness was calculated for both models with the
428
topography observed in 2011. The depth interval observed is the same (2 to 13 m) and the global
429
shape of the bedrock is identical. The main differences are lower than 4 m and are located downhill of
430
the profile, from the abscissa 100 to 230 m where the bedrock topography is more complex. It shows
431
important high frequencies variations that are less pronounced on the topography from Travelletti and
432
Malet (2012).
433
434
To assess the quality of the Q-N algorithm result, we compare the bedrock depth variation with the
435
topography visible on a black and white orthophotograph of 1956 before the landslide (e.g. when the
436
original relief was not covered by the landslide material). The figure 11a shows the orthophotograph
437
of 1956 overlaid on the digital elevation model available for the same date. The seismic profile is also
438
represented. We can notice on the picture the very irregular geometry of the relief composed of
439
alternating crests and gullies of different sizes. The studied profile crosses different geometries. From
440
the point A to B (Fig. 11a), the topography seems first regular, before crossing a large thalweg. After
441
crossing this thalweg, the profile goes across a long transversal crest that forms a bump and then,
442
plunges along the downstream slope. Those patterns are also present in the P-wave tomogram,
443
respectively at 125, 140 m and 160 m.
444
Near the abscissa 180 m, a zone of low velocities (< 900 m.s-1) is observed in depth that can be
445
interpreted as an old small-sized gully, also visible as a black dot on the orthophotograph.
446
447
From this observation, we constructed an interpreted geological model that gathers geophysical and
448
geomorphological information (Fig. 11b). This model is composed of three different materials. The
449
first one is the Callovo-Oxfordian intact black marls that constitute the bedrock, characterized by
450
velocities greater than 3000 m.s-1 (Grandjean et al., 2007) and with a topography following the one
451
visible on the DEM from 1956. The second one is the unconsolidated material that presents P-wave
452
velocities between 400 and 1200 m.s-1, characteristic of unconsolidated deposits (Bell, 2009). It
453
constitutes the active unit of the landslide, e.g. the layer that presents most displacements. The third
454
material, characterized by an interval of velocities between 1500 and 2500 m.s -1 is interpreted as
455
compacted landslide material, because it is only present at a higher depth, where the height of soil
456
above is sufficient to compact the landslide deposit. This layer is not involved in the dynamics of the
457
landslide and is quasi-impermeable (Malet, 2003; Flageollet et al., 2000; Travelletti and Malet, 2012).
458
459
Now that we proved the reliability of the Q-N algorithm tomography to recover the paleotopography,
460
we can assess its contribution by comparing this result from the SIRT tomography of Grandjean and
461
Sage (2004). We can notice on figure 8b and 8c, that the upper part of the profile (between the
462
abscissa 0 and 100 m) presents a smoother contrast between the active unit and the bedrock for the
463
SIRT result. Although two bumps are visible at the abscissa 100 and 140 m, the large in-between
464
depression is not visible. In the lower part, the active layer is deeper than for the Q-N algorithm
465
tomography and presents little lateral heterogeneity that prevent the interpreter from delimitating the
466
landslide layer with accuracy. Globally, the SIRT tomography and the Q-N one are in agreement, but
467
SIRT algorithm does not permit to recover the geotechnical unit limits with the same accuracy and
468
prevents from distinguishing the “Dead body” represented in figure 11b.
469
470
To summarize, we proved that the Q-N algorithm previously developed presented a better horizontal
471
and vertical resolution than the SIRT algorithm (Grandjean and Sage, 2004) and gives the possibility
472
to interpret the bedrock geometry as geomorphological and geological patterns with more details and
473
better resolution than previously. Thanks to the algorithm, we were able to clearly distinguish three of
474
the four geotechnical units proposed by Flageollet et al. (2000). The active unit composed of C1a and
475
C1b geotechnical layers, the “dead body” that represents the C2 layer and the bedrock.
476
477
The seismic attenuation field obtained is in agreement with the P-wave velocities inversed. Although
478
P-wave velocity and seismic attenuation are not linked during the inversion, the attenuation field also
479
permits to distinguish to main materials. The first one presents a relatively low attenuation, that
480
correspond to the bedrock and the “dead body” gathered. The values obtained for this material are
481
around 10-3 Np.m-1, which corresponds to consolidated soils attenuation at 40 Hz (Schön, 1976). The
482
second one is a more attenuating and heterogeneous material that corresponds to the landslide deposit.
483
The attenuation of the landslide layer varies from 4 to 12 10-3 Np.m-1 which are more typical values for
484
unconsolidated soils at this frequency (Schön, 1976). The lateral heterogeneity of the attenuation is
485
important. Areas of high attenuation show attenuation values three times higher than others. The
486
correlation between the SCI and the attenuation is very good, so that we can affirm that attenuation
487
variations are mainly caused by cracking.
488
The seismic wave attenuation tomography highlights two important attenuation zones. The first one, in
489
the upper-part at the abscissas 20-30 seems to be concentrated at the interface between the landslide
490
layer and the bedrock (Fig. 11b). In this area the landslide material undergoes significant shear stresses
491
due to friction between the landslide layer and the bedrock topography. At the surface, cracks wider
492
than 15 cm and measuring several meters in length are visible and the soil is completely remolded.
493
The second zone seems to be associated to the influence of the local bedrock geometry that forms a
494
bump followed by a steep slope (Fig. 11b). Its concave shape is responsible of the tension cracks well
495
visible at the surface and clearly perpendicular to the major change of topography direction (Fig. 3c).
496
Moreover, the soil seems dryer in that zone, certainly because of the geometry of the area that
497
facilitates the drainage of the groundwater table toward lower areas. This phenomenon can even
498
intensify the presence of cracks.
499
We can notice that P-wave velocity and seismic wave attenuation produce complementary results.
500
Only the P-wave velocity tomography can reproduce the sharp geometry of the bedrock topography
501
and differentiate the “dead body” from the bedrock (Fig. 11b). On the other hand, the seismic wave
502
attenuation provides more information on the landslide layer lateral heterogeneity.
503
Here, we can notice that the two high attenuation zones are not correlated with low P-wave
504
velocity as expected (Grandjean et al., 2011). This is explained by the seismic wavepaths going
505
through this weathered material, that are probably too short to impact the velocity, so that only
506
scattering and diffusion effects affect the wave amplitudes.
507
508
Those complementary results highlight the importance of seismic attenuation tomography when
characterizing weathering state of soils in near surface application.
509
It is finally important to note that this geophysical method is, in theory, not adapted to image the
510
near surface zone (first meters in depth). Indeed, the poor Fresnel weighs coverage in that zone
511
prevents to obtain high accuracy results. Moreover, cracks and attenuating objects size is certainly
512
comparable or smaller than the cell size (0.67 m). So, we have to be cautious when interpreting those
513
results because the real attenuation zones could be closer to the surface than those presented on the
514
seismic attenuation tomography.
515
516
5. Conclusions
517
518
We present a P-wave tomography inversion algorithm for highly heterogeneous media which
519
combine improvement of resolution and inversion regularization. We use the Fresnel wavepaths
520
calculated for different sources frequencies to retropropagate the traveltime residuals, assuming that in
521
highly heterogeneous media, the first arrivals are affected by velocity anomalies present in the first
522
Fresnel zone through complex multiple scattering. After verifying the ability of the algorithm to
523
recover complex structures on a synthetic dataset, we apply it on a real dataset acquired on the upper
524
part of the Super-Sauze landslide. We verify that our results are in accordance with previous results
525
from Travelletti and Malet (2012) and dynamics penetrometer tests. It appears that the sharp geometry
526
of the bedrock topography is well recovered. This result is of first importance because bedrock
527
topography is one of the main controlling factors of landslide displacement. Using the wavepaths
528
calculated for P-wave velocity tomography inversion, we invert separately the seismic wave
529
attenuation of the same dataset. Although barely applied in landslide environment, we show that with
530
few efforts it was possible to use the amplitude of the first peak of the wave to get an accurate image
531
of the seismic wave attenuation of the landslide layer. The results are in agreement with the observed
532
surface crack inventory and bring complementary information for the construction of an interpreted
533
model. The use of traveltimes and amplitude of waves permit to create an interpreted geological model
534
that gathers those different details.
535
Such geological models are generally created for hazard assessment through numerical modeling.
536
For this work, the use of geophysical tomography is essential because it is the only tool that can
537
provide continuous and integrated imaging of the soil. In this perspective, our algorithm contribution
538
is to answer one of the major issues raised by Travelletti and Malet (2012), namely, the lack of
539
resolution of geophysical tomography compared to geotechnical tests and geomorphological
540
observations.
541
This is why we believe that the Q-N algorithm could really improve the resolution of the
542
geological model, and thus any numerical hydro-mechanical modeling using this variable as a
543
parameter.
544
545
546
6. Acknowledgements
547
548
This work was supported by the French National Research Agency (ANR) within the Project SISCA
549
“’Système Intégré de Surveillance de Crises de glissements de terrains Argileux, 2009-2012” and the
550
BRGM Carnot institute.
551
552
553
554
555
556
557
558
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Figure Captions
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Figure 1: Fresnel weights (e.g. distance versus depth) computed for a medium characterized by a
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constant velocity of 500 m.s-1 and for three frequencies: a) 100 Hz, b) 200 Hz and c) 300Hz. The
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source is located at x=0 m and the receptor at x=25m, both at 12.5 m of depth.
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Figure 2: Algorithm validation on a synthetic dataset: a) Synthetic initial model, b) Final velocity
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model inverted with the SIRT algorithm, c) Final velocity model inverted with the Q-N algorithm.
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Vertical cross-sections extracted from the three models at a distance of 46 m d), 110 m e) and 145 m
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f).
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Figure 3: Location of the investigated area within the Super-Sauze landslide. The seismic profile
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(red) is represented on an orthophotograph overlaid on a digital elevation model of the landslide. The
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main scarp and the landslide limit. Picture a) shows the seismic device on the field. Picture b) and c)
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represent the observed fissure state of the soil along the profile.
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Figure 4: Data quality: a) Seismic shot# 13. b) Source amplitude spectrum of the total seismic shot
(blue) and source amplitude spectrum of the P-wave of the selected area (red).
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Figure 5: Picking quality: Frequency distribution of differential traveltime errors (e.g. due to
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reciprocal differences). The standard deviation has a value of 2.2047 ms for 427 tested reciprocal
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traveltimes. The grey color indicates the data rejected in the inversion process.
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Figure 6: Example of vertical P-wave velocity profile for the seismic shot #4: a) Slant-stack
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transformation in the t-v (time-velocity) domain for the seismic shot #4, b) Estimated vertical velocity
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profile for each t0, c) Vertical velocity profile inverted for the shot #4.
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Figure 7: Misfit function values for the Q-N and SIRT algorithms.
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Figure 8: Inversion results of the real dataset acquired on the Super-Sauze landslide: a) initial
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velocity model, b) inverted model with the SIRT algorithm of Grandjean and Sage (2004), and c)
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inverted model with the Q-N algorithm.
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Figure 9: Seismic wave attenuation tomography: a) Surface Craking Index (SCI) and b)
attenuation section inverted from the Super-Sauze data.
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Figure 10: Comparison of the bedrock geometry interpreted with the Q-N algorithm and with a
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geological modeler by integration of multi-source information at coarser spatial resolution (Travelletti
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and Malet, 2012) in terms of bedrock depth a) and layer thickness b).
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Figure 11: a) Pre-event topography before the landslide and location of the seismic profile, b)
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Interpreted geological cross-section in 3 layers: the active unit corresponds to the moving landslide
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layer, the dead body is a compacted and quasi-impermeable layer showing low displacements and the
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bedrock constituted of in-situ black marls.